If diffusion between regions of different field strengths is only partial then we might consider a model in which the energy losses of an electron are proportional to
<B^2>_electron = D <B^2> + (1-D) B^2, (12)
which is a combination of the local field B and the average field seen throughout the electron's lifetime. If D=0 then we recover the spectrum described in Section 2, whereas for larger D then the spectrum declines more sharply at high frequencies. Strictly, D should be a function of energy as it depends on both the diffusion time and the energy loss time.
Exactly the same formalism can be applied to a source with a weak magnetic field, so that energy losses are dominated by inverse Compton scattering off the microwave background [this might also be true for radio halos in galaxy clusters (Kim, Tribble & Kronberg 1991), or for sources at high redshift where the energy density of the microwave background is much greater]. In the extreme case the electron energy spectrum is the same everywhere. The radio spectrum differs from place to place because the break frequency B varies. This is equivalent to a case where the electrons sample not only all pitch angles but all field strengths. Models with both synchrotron and inverse Compton losses can be described by equation (12) if we make the identification
D = ^2 / (^2 + <B^2>). (13)
Fig 5. The synchrotron spectrum from electrons in a random magnetic field with pitch angle scattering for various values of the diffusion coefficient D.
I have calculated the spectra for various values of D and show the results in Fig. 5. For small values of D the spectra do not show a sharp break, but this break is apparent for D~>0.2. For D=1 a sharp break is seen, although not quite as sharp as in the standard JP model. Interestingly, for large values of D the spectrum steepens more rapidly than for D=1. This is because a little variation in the energy losses compensates for the factor B in the numerator of , so that varies very little.
If we just consider the effect of inverse Compton losses, then powerful sources with very high field strengths should have radio spectra similar to the KP model. This is indeed the case for Cygnus A (Carilli et al. 1991), where the averaging is mainly along the line of sight as the observations have fairly good resolution. Other sources have equipartition field strengths that are at most a few (eg. 3C234, Alexander 1987), and these sources would be expected to have spectra that are much closer to the JP model. Careful fitting of high frequency spectra could be used to check the reliability of magnetic field strength estimates in radio sources.